Abstract:In this study we construct, in the category XAlg(R) /A of crossed A-modules of R-algebroids, the coproduct of given two crossed A-modules M = (µ : M −→ A) and N = (η : N −→ A) of R-algebroids in two di erent ways: Firstly we construct the coproduct M • * N by using the free product M * N of pre-R-algebroids M and N, and then we construct the coproduct M • N by using the semidirect product M N of M and N via µ. Finally we construct an isomorphism between M • * N and M • N.

In this study, we mainly show that the functor from the category X2Mod of 2‐crossed modules of groups to the category Groups of groups assigning to each 2‐crossed module {}L,M,P,∂2,∂1 the group P, and to each 2‐crossed module morphism ()f2,f1,f0, the group homomorphism f0 is a fibration. In addition, we study some related properties.

Abstract:In this study, we analyse the generators of the Peiffer ideal M, M of a pre-R-algebroid M in a precrossed module M = (µ : M −→ A) in terms of the generators of M for further using and use the outcomes to find the generators of the Peiffer ideal obtained in the coproduct construction of two crossed A-modules of R-algebroids.

In this paper, first, we construct the free modules and precrossed modules of R-algebroids. Then we introduce the Peiffer ideal of a precrossed module and use it to construct the free crossed module.

In this study, given two crossed complexes {\mathcal{C}} and {\mathcal{D}} of associative R-algebras and a crossed complex morphism {f\colon\mathcal{C}\to\mathcal{D}} , we construct a homotopy as a pair {(H,f\/)} , where {H=(H_{n})} is a sequence of R-linear maps {H_{n}\colon C_{n}\to D_{n+1}} . Then we show that for a fixed pair {\mathcal{C}} and {\mathcal{D}} of crossed complexes of associative R-algebras, the family of all homotopies between crossed complex morphisms from {\mathcal{C}} to {\mathcal{D}} has a groupoid structure with crossed complex morphisms as objects and homotopies as morphisms.